What are Sprint/Iteration Gaps?

Sprint gaps give teams the opportunity to recharge and get ready to tackle the next sprint/iteration. In the case that some high priority work can be completed during the sprint gap, teams need to know where closed work will be allocated.
Closed work shows up in whichever sprint the work item it is logged, when the team member closes the work item.
If items have been split, the item and the tasks will be in the new sprint and roll up to the parent item. If an item has not been split, effort will be accounted for in the previous sprint. The effort will show up in whichever sprint the work item it is logged.
Items that are closed during a gap are really no different than any other item being closed outside of a sprint boundary. From a velocity standpoint, it's still assigned to the sprint that it's in. From a Burndown standpoint, it'll be a delta on the gap day. Project/Release Burndown will not be affected. An inline Sprint Burndown (or any burndown filtered to that particular sprint) would not see the post-sprint-end activity.